FOXBOROUGH, Mass. — More than 30 minutes after two long air-horn blasts signaled the end of the Buccaneers’ first joint workout with the Patriots on Tuesday, New England quarterback Tim Tebow was still on the practice fields working to salvage what some believe is a flailing NFL career.

He was on the far sideline near the end zone of the field closest to Gillette Stadium, bunny hopping — first on two legs, then on only his right and finally only on his left. It was not a drill typically assigned to quarterbacks but then again, Tebow is not your typical quarterback.

He has always been a hybrid and it appears the Patriots not only have embraced Tebow’s unique qualities, but also figured out a way to use them.

When an injury to backup Ryan Mallett forced Tebow to take over the Patriots’ offense with 1:22 to play in their preseason opener against the Eagles on Friday, a new offensive attack went in right along with him.

Instead of asking him to drop back into the pocket and look for short, intermediate targets between the numbers and long-range targets outside of them the way Tom Brady and Mallett do, the Patriots turned Tebow loose.

Working at times out of the Pistol formation, which sets the quarterback 4 yards behind center instead of the 7 called for in the shotgun, Tebow ran mostly a read-option attack made up of runs, pitches and short passes.

He didn’t execute it very well, taking three sacks while completing just four of 12 passes for 55 yards, but he ran the ball four times for 31 yards, once again displaying, at the very least, an ability to improvise.

“There were some good things but there were also some things I need to work on,’’ said Tebow, who is fast discovering that the most difficult part of playing for the Patriots might be learning how to play on the fly.

“When you’re called in at a time like that, just physically getting loose is the hardest thing because you’re just standing there for a while,’’ Tebow added. “That’s something I have to practice with and get ready for.’’

Opponents, meanwhile, might have no choice but to get ready for Tebow. Though he is unlikely to replace Mallett as the backup, he might wind up getting his own package of plays to run each week during the regular season.

The concern, of course, is that such a package will require extra practice time to master, which means the Patriots may have to spend less time working on their regular package of plays.

But Patriots coach Bill Belichick has never been afraid to try something new if the end result is an edge over an opponent. The joint practices he agreed to have with the Eagles last week and the Bucs this week are an example of that.

With league rules now limiting the number of workouts teams can have in training camp, joint practices are being seen by some as perhaps the best way for teams to better themselves during the preseason.

So, if the circumstances are right and running Tebow out for a play or two or even a series allows the 36-year-old Brady to get a breather and forces the defense to start scrambling, Belichick might find it beneficial.

But Belichick isn’t quite ready to commit to such a plan. He’s not even ready to commit to Tebow being on his team. When asked about that on Tuesday he said flatly, “We’re not ready to talk about that now.’’

Perhaps, but Tebow does have some potency, particularly in his ability to tuck the ball and run with it. That could aid the Patriots and just happened to come into play during the final drill of Tuesday’s workout.

On the last play of a two-minute drill, Tebow took a snap from center and when the coverage in front of him proved too tight, he rolled out to the right flat, tucked the ball under his arm and won a race to the end zone.

It was vintage Tebow. But so was the extra work he put in after practice and the time, about 20 minutes worth, that he spent signing autographs and posing for pictures after the extra work was done.

After three bumpy years in the league and two teams, nothing about the former University of Florida star — not his work ethic, not his playing style, not even his attitude — has changed.

“My mindset here is just to get better every day, enjoy the process along the way and have fun doing it,’’ Tebow said with a smile. “I’m just blessed to be out here and to have the chance.’’